Classic Geometry Theorems with Complex Numbers

by rrusczyk, Aug 2, 2009, 8:34 PM

I'm considering having a brief section in the Precalc book on classic geometry theorems that can be proved with complex numbers (without too much gory algebra). I have a few of the obvious ones so far, and some not-so-obvious: Pythagorean Theorem, Ptolemy's Inequality, Napoleon, Heron (thanks to Boy Soprano), and the one with three circles and the points determined by the common tangents of each pair being collinear. What are others that would fit nicely here? (I'll be sticking some of the classic triangle theorems, like the Euler line, in a section on vectors and geometry, but go ahead and include those.)

AoPS

math

Comment

4 Comments

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

I think you can do law of cosines (i'm probably wrong)

by Poincare, Aug 2, 2009, 8:50 PM

Report

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

When I took Olympiad Geometry in the spring, I solved a few of the Challenge Problems with complex numbers.

(1) Point $P$ is inside equilateral triangle $ABC$ such that $AP = 4$ , $BP = 3$ , and $CP = 5$ . Find the area of $\triangle ABC$ .

(2) Suppose we invert a segment $AB$ about a circle with center $O$ and radius $r$ . Let the inverse of $AB$ be $A'B'$ . Find a couple of formulas to relate the length of $AB$ to the length of $A'B'$ .

(3) [Variant of IMO 1975 #5] Is there an infinite set of points in the plane such that no three elements of it are collinear, and the distance between any two of them is rational?

Of course, these problems can be solved without complex numbers, but the complex numbers solutions are nice.

(4) On the message board of the class, one student asked if the composition of translations, rotations, and dilations has exactly one fixed point. As long as we avoid obvious degenerate cases, the answer is yes. Complex numbers gives a nice way to handle this question. We can even throw in reflections.

(5) We can prove a few things about the nine-point circle with complex numbers. For example, existence of the circle and its radius.

(6) Zvezda has a handout with a bunch of problems on using complex numbers in geometry. See
http://mathcircle.berkeley.edu/archivedocs/1999_2000/lectures/9900lecturespdf/complexBMC2.pdf.

by Ravi B, Aug 2, 2009, 10:19 PM

Report

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

If you fix the circumcircle of a triangle at the origin, then the formula for the projection of a point on the circle to a chord of the circle is a relatively simple forumula. You can use this to prove simpson line, nine-point circle, and a whole lot of other problems. There is a nice forumula for the area of the pedal triangle of a point ( [ABC]/4R^2| |x|^2-r^2| )

Given a cyclic quadrilateral ABCD, then the orthocenters of ABC, BCD, CDA,DAB form a quadrilateral that is congruent to ABCD

There are a pretty good number of geometric inequalities that can be proven by complex numbers that concern an arbitrary point in the plane and a triangle.

Anything involving similar triangles is easily interpreted in terms of complex numbers. Ex. if ABC, A'B'C' are directly similar triangles, construct points such that AA'A'', BB'B'', CC'C'' are directly similar, then ABC, A'B'C', A''B''C'' are directly similar

by Altheman, Aug 3, 2009, 7:38 PM